Lesson 6 of 7 · 15 min

Sharpe ratio, Treynor ratio, M² and Jensen's alpha

Risk-adjusted performance divides or adjusts excess return by risk: total risk for the Sharpe ratio and M², systematic risk for the Treynor ratio and Jensen's alpha.

In short

  • Performance evaluation has three parts: measurement (return and risk), attribution (sources of performance) and appraisal (skill or luck).
  • Sharpe ratio = excess return ÷ σ: the slope of the CAL. Uses total risk; only meaningful when compared with other Sharpe ratios.
  • Treynor ratio = excess return ÷ β. Uses systematic risk; needs a positive numerator and positive beta.
  • M² = Sharpe ratio × σm\sigma_m + RfR_f: the return the portfolio would earn if levered to market risk. M² alpha = M² − RmR_m. Ranks like Sharpe.
  • Jensen's alpha = Rp−[Rf+βp(Rm−Rf)]R_p - [R_f + \beta_p(R_m - R_f)]: the vertical distance from the SML. Its sign alone shows out- or underperformance.
  • Not fully diversified (e.g. one manager runs everything): use Sharpe/M². Well diversified: use Treynor/Jensen's alpha.

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Sharpe ratio, Treynor ratio, M² and Jensen's alpha · Portfolio Risk and Return: Part II